Small Samples

Distributions

Power

Two-Way Tables

Statisticians use two-way tables and segmented bar charts to
examine the relationship between two categorical variables.

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Two-Way Frequency Tables

Below, the two-way table shows the favorite leisure activities
for 50 adults - 20 men and 30 women. Because entries in the table
are frequency counts, the table is a
frequency
table.

Dance

Sports

TV

Total

Men

2

10

8

20

Women

16

6

8

30

Total

18

16

16

50

Entries in the "Total" row and "Total" column are called
marginal frequencies or the
marginal distribution. Entries in the body
of the table are called joint frequencies.

If we looked only at the marginal frequencies in the Total row,
we might conclude that the three activities had roughly equal appeal.
Yet, the joint frequencies show a strong preference for dance among
women; and little interest in dance among men.

Two-Way Relative Frequency Tables

The table above used frequency counts to describe preferences for leisure activities. Alternatively, we could have used relative frequencies, like percentages or proportions,
to describe the same data.
When we use relative frequencies in a two-way table, table entries are are called
conditional frequencies or the conditional distribution. Here is a version of the leisure-activity table with proportions in the table cells.

Dance

Sports

TV

Total

Men

0.04

0.20

0.16

0.40

Women

0.32

0.12

0.16

0.60

Total

0.36

0.32

0.32

1.00

Relative Frequency for the Whole Table

Two-way tables can show relative frequencies for the
whole table, for rows, or for columns. The table above shows relative frequencies for the whole table.
The following table shows relative frequencies (proportions) for rows.

Dance

Sports

TV

Total

Men

0.10

0.50

0.40

1.00

Women

0.53

0.20

0.27

1.00

Total

0.36

0.32

0.32

1.00

Relative Frequency for Table Rows

And, the next table show relative frequencies (proportions, again) for columns.

Dance

Sports

TV

Total

Men

0.11

0.62

0.50

0.40

Women

0.89

0.38

0.50

0.60

Total

1.00

1.00

1.00

1.00

Relative Frequency for Table Columns

Each type of relative frequency table makes a different contribution to
understanding the relationship between gender and preferences for
leisure activities. For example, the "Relative Frequency for Rows"
table most clearly shows the probability that each gender will
prefer a particular leisure activity. It is easy
to see that the probability that a man will prefer dance is
10%; the probability that a woman will prefer dance is 53%; the
probability that a man will prefer sports is 50%; and so on.

Segmented Bar Charts

Sometimes, relationships are easier to detect when they are
displayed graphically in a segmented bar chart.
A segmented bar chart has one bar for each level of a
categorical variable. Each bar is divided into "segments", such
that the length of each segment indicates proportion or percentage
of observations in a second variable.

The segmented bar chart above uses data from the
"Relative Frequency for Rows" table that we discussed earlier. It shows that women
have an strong preference for dance; while men seldom make dance
their first choice. Men are most likely to prefer sports, but
the degree of male preference for sports over TV is not great.

Test Your Understanding

Problem

A public opinion survey explored the relationship between age
and support for increasing the minimum wage. The results are
summarized below in a two-way frequency table.

For

Against

No opinion

Total

21 - 40

25

20

5

50

41 - 60

20

35

20

75

Over 60

55

15

5

75

Total

100

70

30

200

In the 21 to 40 age group, what percentage supports increasing
the minimum wage?

(A) 12.5%
(B) 20%
(C) 25%
(D) 50%
(E) 75%

Solution

The correct answer is (D). A total of 50 people in the 21 to 40 age
group were surveyed. Of those, 25 were for increasing the
minimum wage. Thus, half of the respondents in the 21 to
50 age group (50%) supported increasing the minimum wage.